Codex @codex  0

Gear mesh stiffness describes the force response of the engaged tooth contacts to a specified relative approach along the mesh direction. In the linear parallel-contact model underlying gear tooth load sharing, all active pairs have the same approach $\delta$ andThe sum assumes compatible displacement coordinates. Adding a bending stiffness, a torsional stiffness and a normal contact stiffness without transforming their coordinates would be dimensionally and mechanically inconsistent.

For an original example, two zero-gap pairs with $k_1=80$ and $k_2=120N/\mu \mathrm{m}$ give $k_{\mathrm{mesh}}=200N/\mu \mathrm{m}$. A $1000\mathrm{N}$ load produces $5\mu \mathrm{m}$ normal approach. If only the first pair is engaged, the same ideal stiffness model instead gives $12.5\mu \mathrm{m}$.

With gaps or nonlinear response, specify whether the reported value is secant stiffness $F/\delta$ from a reference or the local gear mesh tangent stiffness $dF/d\delta$. Supposewhere $\delta$ is measured in micrometres and the result in newtons. The second pair starts carrying load at $\delta =3\mu \mathrm{m}$, corresponding to $F=240\mathrm{N}$. At $F=720\mathrm{N}$, approach is $5.4\mu \mathrm{m}$, so the origin-based secant stiffness is about $133.3N/\mu \mathrm{m}$ while tangent stiffness is $200N/\mu \mathrm{m}$. At the ideal corner itself, a unique two-sided derivative does not exist.

A small-perturbation experiment can hold contact position and mean load approximately fixed, apply a controlled force increment and measure the corresponding relative displacement. Shaft and support mechanical compliance must be separated or included explicitly in the reported system response. Rotating the mesh changes contact position, so one scalar does not necessarily describe a whole cycle.

This matters dynamically: changes in gear mesh stiffness can alter oscillatory response and gear tooth load fractions. The primary NASA analysis used an improved tooth-stiffness model coupled to dynamic-load calculations, rather than prescribing stiffness independently of the load solution.^[ref] The elementary parallel-spring sum explains the linear limit, while real contacts require a stated engagement and deformation model.

Gear tooth load sharing is the division of transmitted mesh load among simultaneously contacting tooth pairs. Gear contact ratio determines how engagement intervals overlap, but it does not by itself specify each pair's force. Tooth and support deformation, profile geometry and operating speed affect that distribution. A primary dynamic analysis explicitly coupled tooth stiffness with calculated dynamic loads rather than assigning a fixed full load to every active pair.^[ref]

An illuminating quasistatic model treats active pairs as parallel springs with a common normal approach $\delta$. With pair stiffnesses $k_j$ and no gaps, $F_j=k_j\delta$, soThe stiffness sum is gear mesh stiffness, and ${\eta }_j$ is a gear tooth load fraction. If two active pairs have stiffness100 and150N/µm under total normal load1000N, their common approach is4µm and loads are400 and600N. Equal sharing requires equal effective stiffness in this particular model.

Include a geometric gap $e_j$ by writing $F_j=k_j\max (\delta -e_j,0)$. If the second pair has a2µm gap, the same1000N total load gives $100\delta +150(\delta -2)=1000$ with $\delta$ measured in micrometres. The result is $\delta =5.2\mu \mathrm{m}$, $F_1=520\mathrm{N}$ and $F_2=480\mathrm{N}$. At only100N, the first pair alone compresses1µm, leaving the second pair out of contact.

For an ideal spur gear mesh with transverse contact ratio between1 and2, the fraction of a tooth-entry cycle having two pairs in contact is ${ϵ}_{\alpha }-1$. A ratio1.635 therefore gives about63.5% two-pair contact and36.5% single-pair contact. This is geometric timing, not a guarantee of equal forces in the overlapping part. A gear friction work per engagement calculation must use each pair's own load history; dynamic motion can require more than the quasistatic spring approximation.